The ISCO should not be confused with the Roche limit, the innermost point where a physical object can orbit before tidal forces break it up.
In general terms, the ISCO will be far closer to the central object than the Roche limit.
In classical mechanics, an orbit is achieved when a test particle's angular momentum is enough to resist the gravity force of the central object.
As the test particle approaches the central object, the required amount of angular momentum grows, due to the inverse square law nature of gravitation.
Orbits can be achieved at any altitude, as there is no upper limit to velocity in classical mechanics.
In GR, gravity is not treated as a central force that pulls on objects; it instead operates by warping spacetime, thus bending the path that any test particle may travel.
The ISCO is the result of an attractive term in the equation representing the energy of a test particle near the central object.
[2] This term cannot be offset by additional angular momentum, and any particle within this radius will spiral into the center.
The precise nature of the term depends on the conditions of the central object (i.e. whether a black hole has angular momentum).
For a non-spinning massive object, where the gravitational field can be expressed with the Schwarzschild metric, the ISCO is located at where
, suggesting that only black holes and neutron stars have innermost stable circular orbits outside of their surfaces.
and the photon sphere so-called unbound orbits are possible which are extremely unstable and which afford a total energy of more than the rest mass at infinity.
This is usually shown by a graph of the orbital effective potential which is lowest at the ISCO.
The equatorial ISCO in the Kerr metric depends on whether the orbit is prograde (negative sign in